1. Field of Invention
This invention pertains to improving the percentage of radiation emitted at visible frequencies from an incandescent emitter, and particularly to providing an incandescent light bulb which significantly improves upon the 7% efficiency currently available from state of the art incandescent light bulbs.
2. Theory
A body which absorbs and emits electromagnetic (E-M) energy is comprised of systems of charged particles which absorb and emit photons and which can, for illustrative purposes, be arranged into groups of many systems with each group forming a volume element Since the systems comprising each volume element mutually interact (primarily via the photon field at incandescent temperatures), thermal equilibrium can be established, at which point there will be an equilibrium distribution of absorption and emission processes and an equilibrium distribution of photons. In addition, to maintain this steady state, the absorption and emission rates (which indicate the degree of interaction among systems) must be equal, and since the energy and particle distributions of an interacting system which is in thermal equilibrium is independent of the degree of interaction, the equilibrium photon distribution is the same function for all materials regardless of the extent of interactions. Plank found the equilibrium photon energy density distribution within volume elements of index of refraction n.sub.r to be equal to (n.sub.r).sup.3 w(v,T)/c where, EQU w(v, T)=8.pi.hv.sup.3 c.sup.-2 (e.sup.hv/kT -1).sup.-1 (1)
Here, h is Plank's constant and k is Boltzman's constant. Therefore, by multiplying the energy density by (c/n.sub.r)a.sub.v, the rate at which radiation of frequency v is being absorbed, the E-M energy per unit time per unit frequency per unit volume absorbed at frequency v in material of absorption coefficient a.sub.v is, a.sub.v (n.sub.r).sup.2 w(v,T). And as stated above, to maintain detailed balance in thermal equilibrium, this must also be the E-M power generation rate.
Consider radiation generated in a horizontal layer, of unit area and of thickness dx, situated a distance x below the surface of a parallel-sided slab of material of thickness X. The radiant power per unit frequency from this layer which reaches the underside of the upper surface at near normal incidence in the solid angle d.omega. is a.sub.v n.sub.r.sup.2 w(v,T)e.sup.-a.sbsp.v.sup.x dxd.omega./4.pi.. Integrating over dx from 0 to X gives the total primary radiation reaching the underside of the surface, and allowing for reflections at the surfaces, the radiant power per unit frequency exiting the surface into the external solid angle d.OMEGA. is .epsilon..sub.v (w(v,T)/4.pi.)d.OMEGA. where .epsilon..sub.v is (1-R)(1-T)/(1RT), and where R is the reflectivity and T, being equal to exp(-a.sub.v X), is the transmissivity of the slab. For a highly absorbing slab (i.e. an optically thick slab where a.sub.v X&gt;&gt;1), .epsilon..sub.v reduces to 1-R while for a highly transmissive slab (i.e. an optically thin slab where a.sub.v X&lt;&lt;1), .epsilon..sub.v reduces to a.sub.v X. A totally absorbing slab (i.e. .epsilon..sub.v equals 1 at all frequencies) is called a black body and w(v,T)/4.pi. is the normal (perpendicular) radiant power density emission per unit frequency of such a body. Allowing for the cosine dependence of non-normal emerging radiation, the radiant power emitted per unit area per unit frequency over all angles in the emergent hemisphere is w(v,T)/4. Correspondingly, the radiant power emitted per unit area per unit wavelength is M.sub.e,.lambda. (T), where .lambda. is the wavelength, and is plotted in FIG. 1 as a function of .lambda. for several values of T. The other curve in the graph, V(.lambda.), is a plot of the relative variation of eye sensitivity with wavelength. Note that the greatest increases, w.r.t. temperature, in the percentage of visible light radiated occur below 2400 K.
In general .epsilon..sub.v can vary from 0 to 1 and is called the spectral emissivity. A body with .epsilon..sub.v constant but less than 1 is called a gray body, and a body with significantly increased emissions (over that of a gray body emitter of the same total power) within a selected spectral bandwidth is called a selective emitter. It is important to note that the derivation of .epsilon..sub.v assumes optical scattering only at the boundary surfaces and is one of the reasons why, for non-scattering media, the only way to obtain a selective emitter is to have a thin body with a.sub.v &gt;&gt;1/X in the spectral region of interest (i.e. at visible frequencies) only, or a thick body with R small in the spectral region of interest. However, the costs and technical problems associated with fabricating macroscopic refractory bodies with negligible internal optical scattering renders these types of selective emitters impractical. Moreover, the physical support needed for a large surface thin body, coupled with its IR transparency means the substrate generated IR is externally transmitted, thereby negating the thin body selectivity. And the relatively small variations in R exhibited by most refractories within the visible and near IR regions is not enough to effect an efficient thick body selective emitter.
For inhomogeneous media with scattering throughout the body, the calculation of .epsilon..sub.v is much more involved because the necessary radiant energy transfer relationships are in terms of the integro-differential equation, ##EQU1## i.sub.v (x,.omega.) is the intensity at x in the direction .omega.,a.sub.v is the absorption coefficient at v, i.sub.vb is the black body radiant power density at v at the local temperature, .sigma..sub.v is the scattering coefficient, and .phi. is the scattering function that tells how much of the intensity in direction .omega..sub.i is scattered into direction .omega.. Chub and Lowe (1993) solved this equation for a mono-temperature slab of thickness d, index of refraction n.sub.f, and surface reflectivity .rho..sub.v0, which is uniformly imbedded with isotropically scattering optical inhomogeneities, and which is mounted on a substrate of surface reflectivity .rho..sub.vs (w.r.t. the slab) and emissivity .epsilon..sub.vs (w.r.t. the slab) to obtain the emissivity (where .epsilon..sub.v is equivalently defined as the intensity integrated over all angles within the emergent hemisphere divided by that of a black body at the same temperature) as a function of optical thickness, scattering and absorption. ##EQU2## Definitions of the various functions are given in Appendix A. A plot of .epsilon..sub.v as a function of z.sub.v for asymptotically large values of K.sub.vd (i.e. K.sub.vd &gt;&gt;1), where z.sub.v =.sigma..sub.v /(a.sub.v +.sigma..sub.v) and K.sub.vd =(a.sub.v +.sigma..sub.v)d, is shown in FIG. 2. The constants used for the slab and substrate, n.sub.f =2, .rho..sub.v0 =0.15, .epsilon..sub.vs =1, .rho..sub.vs =0, reflect the optical characteristics of the emitter. It is clear from the figure that, with scattering, highly discriminating selective emitters can be had for optically thick media with .epsilon..sub.v large in the spectral region of interest.